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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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23 Aug 2015, 22:49
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59% (01:30) correct 41% (01:15) wrong based on 517 sessions
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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23 Aug 2015, 23:09
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
1/2 * (2^30 + 2^30) = 2^30
2^29 + 2^29 = 2^30
The above equation is similar to 2^x + 2^y =2^30. so x =29 and y=29
Ans:D



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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23 Aug 2015, 23:22
Bunuel wrote: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
A. 30 B. 32 C. 46 D. 58 E. 64 Ans: D Solution: 2^x + 2^y =2^30 there must be some value of as we know 2^30 only has 2 as its factor so if i take anything common from LHS of the equation the remaining part must be in form of 2 only. I can take common if and only if x=y; otherwise the remaining part inside the bracket will become odd for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y so now=> the equation becomes 2^x + 2^x =2^30 because x=y 2^x (1+1)= 2^30 2^x * 2 = 2^30 x+1 = 30 x= 29 and y = 29 x+y = 58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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24 Aug 2015, 00:48
Bunuel wrote: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. IMO: D Both sides consists of only powers of 2. Thus in order to that happen x = y must be the case if x=y then \(2^x +2^y =2 ^30\) ==> \(2^x +2^x =2 ^30\) \(2(2^x) =2 ^30\) x+1 =30 x=29 Thus x+y = 29+29 =58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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24 Aug 2015, 21:38
\(2^x+2^y\,=\,2^{30}\) \(x+y\,=\,?\)
\(2^x\,(1+2^{yx})\,=\,2^{30}\) \(1+2^{yx}\,=\,2^{30x}\) \(1\,=\,2^{30x}2^{yx}\)
Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero i.e. \(2^1\,\,2^0\,=\,1\)
> \(30x\,=\,1\,\,\,\) and \(\,\,\,yx\,=\,0\) \(x\,=\,29\,\,\,\) and \(\,\,\,x\,=\,y\) \(x+y\,=\,58\)
Answer D



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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25 Aug 2015, 01:43
2^1 = 2^0 + 2^0 2^2 = 2^1 + 2^1 2^3 = 2^2 + 2^2 . . . 2^30 = 2^29 + 2^29
x + y = 58
Option D



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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25 Aug 2015, 12:15
Bunuel wrote: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. 2^x+2^y=2^30 Now, 2^30=2^29+2^29=2*2^29 or x+y=29+29=58 Answer D



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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25 Aug 2015, 14:11
dkumar2012 wrote: for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y This is a great solution. A little more detail for those wanting to see why all cases will not work (not just the one above) 2^x + 2^y = 2^x * (1+2^(yx)) = 2^30 This means that 1+2^(yx) has to be a multiple of 2 and this can only be achieved if 2^(yx) = 1 = 2^0.



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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30 Aug 2015, 08:41
Bunuel wrote: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. VERITAS PREP OFFICIAL SOLUTION:This problem is a good candidate for testing small numbers to find patterns. While your instincts may tell you to simply set x + y equal to 30, small numbers will show that you cannot simply set them equal. If you try \(2^x+2^y=2^6\), for example you can't find values for x + y = 6 that will set the sum equal to 64. The powers of 2 are: 2 4 8 16 32 64 The only pairing that will work is 32 + 32 = 64, meaning that you need \(2^5+2^5=2^6\), which should make sense: 2 times 2^5 will equal 2^6. So this example should teach you that you need to add two \(2^{29}\)s together to get to \(2^{30}\). So x and y are each 29, making the sum x + y equal to 58, answer choice D.
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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19 Nov 2015, 17:58
Bunuel wrote: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. 2^x+2^Y = 2^30 2^x30 + 2^y30 = 1 (Divide LHS and RHS by 2^30) 1/2+1/2 = 1 Therefore 2^x30=1/2 i.e. x30 = 1 hence X= 29 similarly y30 = 1 and y = 29 X+Y = 29+29 = 58 Ans D



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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08 Jan 2016, 10:08
shasadou wrote: For integers x and y, 2^x+2^y=2^30. What is the value of x+y?
A. 30 B. 32 C. 46 D. 58 E. 64 You are given 2^x+2^y=2^30 > in order to understand the question, try to see what values can y take if you start with fixed values for x > remember that x and y MUST be integers. Thus, lets have x =1 > \(2^y=2^{30}2\) > y can not be an integer. Try a bigger value for x. x = 15 > \(2^y=2^{30}2^{15}\) > \(2^y=2^{15}(2^{15}1)\) , keep trying and you will see that there will always be a 1 inside the bracket except when you write the given expression as \(2^x+2^y=2^{30}\) >\(2^x+2^y=2*2^{29}\) > \(2^x+2^y=2^{29}+2^{29}\) > compare both sides of the equation to get x=y=29 and x+y = 58. D is thus the correct answer. Hope this helps.



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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08 Jan 2016, 23:07
shasadou wrote: For integers x and y, 2^x+2^y=2^30. What is the value of x+y?
A. 30 B. 32 C. 46 D. 58 E. 64 This is a property of exponents: \(2^1 + 2^1 = 2^2\) \(2^2 + 2^2 = 2^3\) (because \(2^2 * (1 + 1) = 2^2 * 2\)) Similarly, \(2^3 + 2^3 = 2^4\) \(2^4 + 2^4 = 2^5\) etc So \(2^{29} + 2^{29} = 2^{30}\) \(x + y = 29 + 29 = 58\) Similarly, \(3^1 + 3^1 + 3^1 = 3^2\) \(3^2 + 3^2 + 3^2 = 3^3\) \(3^3 + 3^3 + 3^3 = 3^4\) and so on...
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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11 Jan 2016, 09:12
shasadou wrote: For integers x and y, 2^x+2^y=2^30. What is the value of x+y?
A. 30 B. 32 C. 46 D. 58 E. 64 Hi, here we are adding two different numbers which are power to base 2 resulting into another number which has a base of 2.. the logic is .. if we are adding 2^x and 2^y to get 2^z, only way is to get 2 out of the addition and that is possible only when x and y are same.. so 2^x+2^y=2^30..where x =ycan be written as 2^x+2^x=2^30 2*2^x=2^30.. x+1=30 ..x=29.. so x+y=29+29=58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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16 Apr 2016, 19:28
Here's a better way to solve the problem (suggested by Anthony Ritz  big thank you! ). Step 1: factor out the given equation and see if there is a pattern\(2^x(1 + 2^{yx}) = 2^{30}\) Hm, interesting. We know that \(2^x\) is always even, so \((1 + 2^{yx})\) should also be even since \(2^{30}\) is even. For \((1 + 2^{yx})\) to be an even integer, \(2^{yx}\) needs to be odd. What situation will \(2^{yx}\) be an odd number? When \(2^{yx} = 1\)! Hence,\(yx = 0\) and consequently \(x=y\). Step 2: plug in our findings to the equation\(2^x + 2^y = 2^x + 2^x = 2 * 2^x = 2^{1+x} = 2^{30}\) The last two parts of our equation tells us that \(1+x = 30\). In conclusion, \(x = y = 29\). Answer: \(x + y = 58\).



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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17 Apr 2016, 03:42
It's actually really easy to spot a pattern here. 2^2 + 2^2 = 8 2^3 = 8 2^4 + 2^4= 16 + 16 = 32 2^5 = 32 So 2^x +2^y = 2^n to get x+y you just do 2(n1).
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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05 Nov 2016, 18:35
I used a simple tester example to see what would happen if I wanted to find a smaller exponential value of 2.
I used the following:
2^4 = 2^x + 2^y > 8+8 (i.e. 2^3 + 2^3) satisfies this...
sum of x and y has to be one less than exponent we are looking for. Thus 29+29 = 58 and we have our answer.



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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05 Nov 2016, 23:34
Bunuel wrote: For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. Love this as a 700 level question. Easy if you keep your head on straight. Plug in 2^2 + 2^2= 8 = 2^3.... Plug in 2^3 +2^3= 16 = 2^4....As we can see the pattern shows that 2^29 + 2 ^29 will equal 2^30. Then we can add 29 + 29 and get 58. Under 30 seconds oooooooooooooooooooooh yea



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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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11 May 2018, 05:32
VeritasPrepKarishma wrote: shasadou wrote: For integers x and y, 2^x+2^y=2^30. What is the value of x+y?
A. 30 B. 32 C. 46 D. 58 E. 64 This is a property of exponents: \(2^1 + 2^1 = 2^2\) \(2^2 + 2^2 = 2^3\) (because \(2^2 * (1 + 1) = 2^2 * 2\)) Similarly, \(2^3 + 2^3 = 2^4\) \(2^4 + 2^4 = 2^5\) etc So \(2^{29} + 2^{29} = 2^{30}\) \(x + y = 29 + 29 = 58\) Similarly, \(3^1 + 3^1 + 3^1 = 3^2\) \(3^2 + 3^2 + 3^2 = 3^3\) \(3^3 + 3^3 + 3^3 = 3^4\) and so on... Thanks for making things much easier!! Love your blogs as well. Keep up the great work.



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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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11 May 2018, 13:42
Alternative approach if you don's see the pattern: \(2^x + 2^y=2^{30}\), where \(2^x + 2^y\)=\((2^x +2^y)^22*2^{x+y}=2^{30}\), where \((2^x + 2^y)^2=2^{60}\) \(2^{30}*(2^{30}1)=2^{x+y+1}\) , we can omit 1 since it almost doesn't change the result, so \(2^{60}= 2^{x+y+1}\), x+y+1=60. x+y = 59  yes I know that this is not 58, but this is the closest result. Answer (D)
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]
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08 Jun 2018, 02:06
Bunuel wrote: For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?
A. 30 B. 32 C. 46 D. 58 E. 64
Kudos for a correct solution. \(2^x + 2^y=2^{30}\) which can be written as, \(2^{29}*({2^{x29}} + {2^{y29}})=2^{30}\) Hence, \({2^{x29}} + {2^{y29}} = 2\) Since x & y are integers, we got \(x29 = 0\) & \(y29 = 0\) \(x = 29\), \(y =29\), \(x+y = 58\) Answer D. Thanks, GyM




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